In 1969, S. Tanno classified connected almost contact metric manifolds whose automorphism groups possess the maximum dimension [14]. For such a manifold, the sectional curvature of plane sections containing ξ is a constant, say c. He showed that they can be divided into three classes: (1) homogeneous normal contact Riemannian manifolds with c > 0, (2) global Riemannian products of a line or a circle with a Kaehler manifold of constant holomorphic sectional curvature if c = 0 and (3) a warped product space R ×_fC if c > 0. It is known that the manifolds of class (1) are characterized by admitting a Sasakian structure. K. Kenmotsu [12] characterized the differential geometric properties of the manifolds of class (3); the structure so obtained is now known as Kenmotsu structure. Kenmotsu manifolds have been studied by various authors such as G. Pathak and U. C. De [13], J. B. Jun, U. C. De and G. Pathak [11], A. Yildiz et al. [17] and many others.

In 1924, the idea of semi-symmetric linear connection on a differentiable manifold was introduced by A. Friedmann and J. A. Schouten [8]. In 1930, E. Bartolotti [2] gave a geometrical meaning of such a connection. In 1932, H. A. Hayden [10] introduced semi-symmetric metric connection in Riemannian manifolds and this was studied systematically by K. Yano [15]. The semi-symmetric non-metric connection in a Riemannian manifold have been studied by N. S. Agashe and M. R. Chafle [1], U. C. De and S. C. Biswas [7], L. S. Das, M. Ahmad and A. Haseeb [6], S. K. Chaubey and R. H. Ojha [5] and others. Recently, A. Haseeb, M. A. Khan and M. D. Siddiqi studied an ε-Kenmotsu manifold with a semi-symmetric metric connection and obtained some new noteworthy results on this manifold [9].

Let ∇ be a linear connection in an n-dimensional differentiable manifold M. The torsion tensor T and the curvature tensor R of ∇ are given respectively by

The connection ∇ is said to be symmetric if its torsion tensor T vanishes, otherwise it is non-symmetric. The connection ∇ is said to be metric connection if there is a Riemannian metric g in M such that ∇g = 0, otherwise it is non-metric. It is well known that a linear connection is symmetric and metric if it is the Levi-Civita connection.

A linear connection ∇ is said to be semi-symmetric connection if its torsion tensor T is of the form

where η is a 1-form.

Semi-symmetric connections play an important role in the study of Riemannian manifolds. There are various physical problems involving the semi-symmetric metric connection. For example, if a man is moving on the surface of the earth always facing one definite point, say Jaruselam or Mekka or the North pole, then this displacement is semi-symmetric and metric [8].

Motivated by the above studies, in this paper we obtain some new results on concircular curvature tensor in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection. The paper is organized as follows : In Section 2, we give a brief introduction of a Kenmotsu manifold and define semi-symmetric non-metric connection. In Section 3, we find the curvature tensor, Ricci tensor and scalar curvature in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection. Section 4 deals with the study of concircular curvature tensor in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection and we establish the relation between concircular curvature tensors with respect to the connections ∇ and ∇̄. In Section 5, we show that the Kenmotsu manifold satisfying the condition C̄^*(ξ, X) S̄ = 0 is an η-Einstein manifold. Section 6 is devoted to the study of concircularly flat and ξ-concircularly flat Kenmotsu manifolds with respect to the connection ∇̄. In Section 7, we study concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection.

An n-dimensional smooth manifold (M, g) (n = 2m + 1 > 1) is said to be an almost contact metric manifold [3], if it admits a (1, 1) tensor field φ, a structure vector field ξ, a 1-form η and the Riemannian metric g which satisfy

for arbitrary vector fields X and Y on M.

An almost contact metric manifold M(φ, ξ, η, g) is said to be Kenmotsu manifold if ([12])

where ∇ denotes the Riemannian connection of g.

Also the following relations hold in a Kenmotsu manifold [12]:

where g(QX, Y ) = S(X, Y ). It yields to

Example

We consider the three dimensional manifold M = [(x, y, z) ∈ R³ | z ≠ 0], where (x, y, z) are the cartesian coordinates in R³. Choosing the vector fields

which are linearly independent at each point of M. Let g be the Riemannian metric define by

Let η be the 1-form defined by η(Z) = g(Z, e₃) for any vector field Z on TM. Let φ be the (1, 1) tensor field defined by φ(e₁) = −e₂, φ(e₂) = e₁, φ(e₃) = 0. Then by the linearity property of φ and g, we have

for any vector fields Z, W on M.

Let ∇ be the Levi-Civita connection with respect to the metric g. Then we have

The Riemannian connection ∇ with respect to the metric g is given by

From above equation which is known as Koszul’s formula, we have

Using the above relations, for any vector field X on M, we have

for ξ ∈ e₃. Hence the manifold M under consideration is a Kenmotsu manifold of dimension three.

Definition 2.1([16])

A Kenmotsu manifold M is said to be an η-Einstein manifold if its Ricci tensor S of type (0,2) is of the form

where a and b are smooth functions on M. If a = 0, then the manifold will be called a special type of η-Einstein manifold.

Definition 2.2([4])

The concircular curvature tensor C^* in a Kenmotsu manifold M is defined by

where R and r are the Riemannian curvature tensor and the scalar curvature of the manifold respectively.

Definition 2.3

A Kenmotsu manifold M is said to be symmetric if

for all vector fields X, Y, Z, W on M, where R is the curvature tensor with respect to ∇.

Definition 2.4

A Kenmotsu manifold M is said to be concircularly symmetric if

for all vector fields X, Y, Z, W on M, where C^* is the concircular curvature tensor defined by (2.14).

A linear connection ∇̄ in M is called semi-symmetric connection if its torsion tensor

satisfies

Further, a semi-symmetric connection is called semi-symmetric non-metric connection [1] if

Let M be an n-dimensional Kenmotsu manifold and ∇ be the Levi-Civita connection on M, the semi-symmetric non-metric connection ∇̄ on M is given by

The curvature tensor R̄ of semi-symmetric non-metric connection ∇̄ in M is defined by

Using Equations (2.6) and (2.20) in (3.1), we get

where

is the Riemannian curvature tensor of the connection ∇.

Contracting X in (3.2), we get

where S̄ and S are the Ricci tensors of the connections ∇̄ and ∇, respectively on M.

This gives

Contracting again Y and Z in (3.3), it follows that

where r̄ and r are the scalar curvatures of the connections ∇̄ and ∇, respectively on M.

Lemma 3.1

Let M be an n-dimensional Kenmotsu manifold with respect to the semi-symmetric non-metric connection. Then

Theorem 3.2

In a Kenmotsu manifold with respect to the semi-symmetric non-metric connection the curvature condition R̄(ξ, Y ). S̄ = 0 is satisfied.

Analogous to the Definition 2.2, the concircular curvature tensor C̄^* on a Kenmotsu manifold with respect to the semi-symmetric non-metric connection is given by

where R̄ and r̄ are the curvature tensor and scalar curvature with respect to the connection ∇̄, respectively.

By making use of (3.2) and (3.5) in the above equation, we obtain

where C^*(X, Y )Z is the concircular curvature tensor with respect to the connection ∇ and is given by (2.14). The equation (4.2) is the relation between the concircular curvature tensors with respect to the connections ∇̄ and ∇.

Interchanging X and Y in (4.2), we have

On adding (4.2) and (4.3) and using the fact that R(X, Y )Z + R(Y, X)Z = 0, we get

From (3.2), (4.1) and first Bianchi identity R(X, Y )Z +R(Y, Z)X +R(Z, X)Y = 0 with respect to ∇, we obtain

Equation (4.4) (resp.,(4.5)) shows that in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection the concircular curvature tensor is skew-symmetric (resp., cyclic).

Next, taking inner product of (4.1) with ξ, we have

Using (3.2) and (3.5) in (4.6), we find

C̄^*(ξ, X). S̄ = 0

Now we consider a Kenmotsu manifold with respect to the semi-symmetric non-metric connection ∇̄ satisfying the condition

Then we have

By virtue of (2.2), (3.5), (3.7) and (4.1), we have

From (5.2), (5.3) and (5.4), we have

which on using (3.8), reduces to

By using (3.3), (5.5) takes the form

This implies that either the scalar curvature of M is r = n² − 3n + 2 or S = (n − 1)g(X, Y ) − 2(n − 1)η(X)η(Y ). The converse part is trivial.

Thus we can state the following theorem.

Theorem 5.1

The concircular curvature tensor with respect to the semi-symmetric non-metric connection in a Kenmotsu manifold satisfies C̄^*(ξ, X).S̄ = 0 if and only if either M has the scalar curvature n² − 3n + 2 or M is an η-Einstein manifold.

Firstly, we assume that the manifold M with respect to the semi-symmetric non-metric connection is concircularly flat, that is, C̄^*(X, Y )Z = 0. Then from (4.1), it follows that

Taking inner product of the above equation with ξ, we have

In view of (2.2), (2.9), (3.2) and (3.5), we get

This implies that either the scalar curvature of M is r = −(n − 1)(n + 2) or

Putting Y = ξ in (6.4) and using (2.2), it follows that

Taking X = QX in (6.5), and using (2.10), we get

Hence we can state the following theorem.

Theorem 6.1

For a concircularly flat Kenmotsu manifold with respect to the semi-symmetric non-metric, either the scalar curvature is −(n − 1)(n + 2) or the manifold is a special type of η-Einstein manifold.

Secondly, we assume that the manifold M with respect to the semi-symmetric non-metric connection is ξ-concircularly flat, that is, C̄^*(X, Y )ξ = 0. For this taking Z = ξ in (6.1), we have

In view of (3.5) and (3.6), we have

Taking Y = ξ in (6.8) and using (2.2), we get

Taking inner product of (6.9) with U, we have

Now taking X = QX in (6.10), we have

By making use (2.10) and the fact that S(X, U) = g(QX, U), we find

This implies that either the scalar curvature of M is r = (n − 1)(n − 2) or

Hence we can state the following theorem.

Theorem 6.2

For a ξ-concircularly flat Kenmotsu manifold with respect to the semi-symmetric non-metric, either the scalar curvature is (n − 1)(n − 2) or the manifold is a special type of η-Einstein manifold.

Next, taking Z = ξ in (4.2) and using (2.2), we get

Making use of (2.2), it follows that

Since η(X)Y − η(Y )X = R(X, Y )ξ ≠ 0, in a Kenmotsu manifold, in general, then we have the following theorem.

Theorem 6.3

ξ-concircularly flatness in Kenmotsu manifolds with respect to the semi-symmetric non-metric connection and the Levi-Civita connection are not equivalent.

Using (2.14), we can write

Now differentiating (4.2) with respect to W, we have

Making use of (2.6) in (7.2), we obtain

From (7.1) and (7.3), we get

In view of (4.1) and (4.7), (7.4) takes the form

Now differentiating (2.14) with respect to W, we get

From (7.5) and (7.6), we obtain

from which we have

where

If the scalar curvature r is constant and F(X, Y, Z, W) = 0, then (7.8) reduces to

Hence we can state the following theorem.

Theorem 7.1

A Kenmotsu manifold is concircularly symmetric with respect to the semi-symmetric non-metric connection ∇̄ if and only if it is symmetric with respect to Riemannian connection ∇, provided the scalar curvature r is constant and F(X, Y, Z, W) = 0.

The author wishes to express his sincere thanks and gratitude to the referee for valuable suggestions towards the improvement of the paper.